Exercise 1914Calculus › Multivariable calculus › Vector calculusL1Added 3 Nov 2026

Green's theorem with a general field

#Green#general field#region

Statement

Compute $\oint_C(x^2-y^3)\,dx+(x^3+y^2)\,dy$ where $C$ is the circle $x^2+y^2=R^2$ (counterclockwise).

Hint

$Q_x-P_y=3x^2+3y^2$ .

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Identify. $P=x^2-y^3$ , $Q=x^3+y^2$ . Then $Q_x-P_y=3x^2+3y^2$ .

Green. $\oint=3\iint_D(x^2+y^2)\,dx\,dy$ .

Polar. $3\int_0^{2\pi}\int_0^Rr^3\,dr\,d\theta=3\cdot 2\pi\cdot\frac{R^4}{4}=\frac{3\pi R^4}{2}$ .

Conclusion. $\oint=\frac{3\pi R^4}{2}$ .